Is there anything preventing researchers from founding new research institutions located anywhere, providing them with the flexibility of working in their desired location and still enabling them to get their research and funding from external grants?
Let's take as an example the US National Science Foundation. PAPPG I.E, "Who May Submit Proposals", lists:
(a) Institutions of Higher Education (IHEs) — Two- and four-year IHEs (including community colleges) accredited in, and having a campus located in the U.S., acting on behalf of their faculty members. [...]
(b) Non-profit, Non-academic Organizations — Independent museums, observatories, research laboratories, professional societies, and similar organizations located in the U.S. that are directly associated with educational or research activities.
(c) Tribal Governments [...]
as well as, when specifically stated as eligible for a program: for-profit organizations, state and local governments, and foreign organizations ("rarely").
You'd presumably be trying to qualify under (b). There isn't any more specific definition as far as I can tell, so it'd be up to you to convince the agency officials that your organization falls in this category. But the examples given suggest that they have in mind firmly established institutions.
Moreover, there is an explicit provision that "unaffiliated individuals" are not eligible. So you'd also have to convince them that your organization isn't just a shell for an unaffiliated individual, which in fact is exactly what you are trying to create. It'd probably be rather obvious that it is, if for instance the organization has only one employee (you).
Supposing that you manage to clear that hurdle, the next one is 2 CFR 200 which gives criteria for federal grants. 200.206 lists risks that the granting agency must evaluate:
(i) Financial stability. Financial stability;
(ii) Management systems and standards. Quality of management systems
and ability to meet the management standards prescribed in this part;
(iii) History of performance. The applicant's record in managing
Federal awards, if it is a prior recipient of Federal awards,
including timeliness of compliance with applicable reporting
requirements, conformance to the terms and conditions of previous
Federal awards, and if applicable, the extent to which any previously
awarded amounts will be expended prior to future awards;
(iv) Audit reports and findings. Reports and findings from audits
performed under subpart F of this part or the reports and findings of
any other available audits; and
(v) Ability to effectively implement requirements. The applicant's
ability to effectively implement statutory, regulatory, or other
requirements imposed on non-Federal entities.
You would not score well on these compared to a large, established institution.
As to the proposal itself, the NSF Merit Review Principles include:
Are there adequate resources available to the PI (either at the home organization or through collaborations) to carry out the proposed activities?
"Resources" includes not only things like scientific equipment, but also basic stuff like office space, IT services, accounting, human resources management, compliance staff, legal support, etc. For established institutions, it's usually taken for granted that they can reliably provide these, but your shell institution would not find it so easy to assure them of this.
If you did receive an award, you'd be responsible for complying with 2 CFR 200 Subpart D, Post Federal Award Requirements. 200.302 in particular requires your organization to have a fairly elaborate financial management infrastructure. 200.303 requires internal controls, which you probably don't have if the only person supervising your activities is you.
So in short: No, you can't feasibly set up on your own an organization that would be eligible to receive federal grants, reasonably competitive in applying for them, and able to properly administer them if received.